Fifth Property of the Euclidean Metric - Revision history

Ranjelin at 07:00, 23 September 2016

2016-09-23T07:00:39Z

←Older revision		Revision as of 07:00, 23 September 2016
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	namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>		namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>
-	* <math>\sqrt{d_{ij}}\geq0\,,~~i\~~neq~~ j</math>                                       '''('''nonnegativity''')'''	+	* <math>\sqrt{d_{ij}}\geq0\,,~~i\not= j</math>                                       '''('''nonnegativity''')'''
	* <math>\sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j</math>                            '''('''self-distance''')'''		* <math>\sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j</math>                            '''('''self-distance''')'''
	* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math>                                                  '''('''symmetry''')'''		* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math>                                                  '''('''symmetry''')'''
-	* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\~~neq~~\!j\!\~~neq~~\!k</math>    '''('''triangle inequality''')'''	+	* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\not=\!j\!\not=\!k</math>    '''('''triangle inequality''')'''


	==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''==		==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''==
	Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,		Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,
-	for all  <math>i_{},j_{},\ell\~~neq~~ k_{}\!\in\!\{1\ldots_{}N\}</math> ,	+	for all  <math>i_{},j_{},\ell\not= k_{}\!\in\!\{1\ldots_{}N\}</math> ,
	<math>i\!<\!j\!<\!\ell</math> ,  and for  <math>N\!\geq_{\!}4</math>  distinct points  <math>\,\{x_k\}\,</math> ,  the inequalities		<math>i\!<\!j\!<\!\ell</math> ,  and for  <math>N\!\geq_{\!}4</math>  distinct points  <math>\,\{x_k\}\,</math> ,  the inequalities

Ranjelin at 06:56, 23 September 2016

2016-09-23T06:56:50Z

←Older revision		Revision as of 06:56, 23 September 2016
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	For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>\,x_i\,</math> and <math>\,x_j\,</math> is defined		For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>\,x_i\,</math> and <math>\,x_j\,</math> is defined

-	<math>~~\begin{array}{rl}~~d_{ij}	+	<math>d_{ij}=\|\|x_i-x_j\|\|^2
-	~~\!\!&~~=~~\,\~~\|x_i-~~_{}~~x_j\\|^2	+	=(x_i-x_j)^{\rm T}(x_i-x_j)=\|\|x_i\|\|^2+\|\|x_j\|\|^2-2x^{\rm T}_ix_j\\\\
-	~=~(x_i-~~_{}~~x_j)^{\rm T}(x_i-~~_{}~~x_j)~=~\\|x_i\\|^2+\\|x_j\\|^2-~~2_{}x~~^{\rm T}~~_i\!x_j~~\\\\	+	=\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right]
-	&=\,\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\end{array}\right]	+	\left[\begin{array}{cc}x_i\\x_j\end{array}\right]</math>
-	\left[~~\!\!~~\begin{array}{cc}x_i\\x_j\end{array}~~\!\!~~\right]	+
-	~~\end{array}~~</math>	+

	Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [[http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Dattorro, ch.5.2]]		Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [[http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Dattorro, ch.5.2]]

Ranjelin at 05:35, 5 December 2011

2011-12-05T05:35:12Z

←Older revision		Revision as of 05:35, 5 December 2011
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	<math>\begin{array}{rl}d_{ij}		<math>\begin{array}{rl}d_{ij}
	\!\!&=\,\\|x_i-_{}x_j\\|^2		\!\!&=\,\\|x_i-_{}x_j\\|^2
-	~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\\|x_i\\|^2+\\|x_j\\|^2-2_{}x^~~T_i~~\!x_j\\\\	+	~=~(x_i-_{}x_j)^{\rm T}(x_i-_{}x_j)~=~\\|x_i\\|^2+\\|x_j\\|^2-2_{}x^{\rm T}_i\!x_j\\\\
-	&=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\end{array}\right]	+	&=\,\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\end{array}\right]
	\left[\!\!\begin{array}{cc}x_i\\x_j\end{array}\!\!\right]		\left[\!\!\begin{array}{cc}x_i\\x_j\end{array}\!\!\right]
	\end{array}</math>		\end{array}</math>

Ranjelin at 21:07, 19 November 2011

2011-11-19T21:07:02Z

←Older revision		Revision as of 21:07, 19 November 2011
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	\!\!&=\,\\|x_i-_{}x_j\\|^2		\!\!&=\,\\|x_i-_{}x_j\\|^2
	~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\\|x_i\\|^2+\\|x_j\\|^2-2_{}x^T_i\!x_j\\\\		~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\\|x_i\\|^2+\\|x_j\\|^2-2_{}x^T_i\!x_j\\\\
-	&=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{*{20}r}\!I&-I\\\!-I&I\end{array}\right]	+	&=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\end{array}\right]
-	\left[\!\!\begin{array}{*{20}c}x_i\\x_j\end{array}\!\!\right]	+	\left[\!\!\begin{array}{cc}x_i\\x_j\end{array}\!\!\right]
	\end{array}</math>		\end{array}</math>

Dattorro: Protected "Fifth Property of the Euclidean Metric" [edit=autoconfirmed:move=autoconfirmed]

2009-01-10T09:28:36Z

Protected "Fifth Property of the Euclidean Metric" [edit=autoconfirmed:move=autoconfirmed]

←Older revision

Revision as of 09:28, 10 January 2009

Dattorro at 09:23, 10 January 2009

2009-01-10T09:23:21Z

←Older revision		Revision as of 09:23, 10 January 2009
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	namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>		namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>
	* <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math>                                       '''('''nonnegativity''')'''		* <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math>                                       '''('''nonnegativity''')'''
-	* <math>\sqrt{d_{ij}}=0~~\,,~~~~i=j</math> ~~           ~~                           '''('''self-distance''')'''	+	* <math>\sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j</math>                            '''('''self-distance''')'''
	* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math>                                                  '''('''symmetry''')'''		* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math>                                                  '''('''symmetry''')'''
	* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math>    '''('''triangle inequality''')'''		* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math>    '''('''triangle inequality''')'''

Dattorro at 09:18, 10 January 2009

2009-01-10T09:18:08Z

←Older revision		Revision as of 09:18, 10 January 2009
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	== References ==		== References ==
-	* Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry], Meboo, ~~2007~~	+	* Dattorro, [http://www.meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], Meboo, 2005

Cslaw at 19:39, 26 July 2008

2008-07-26T19:39:26Z

←Older revision		Revision as of 19:39, 26 July 2008
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	[[Image:Thefifth.jpg\|thumb\|right\|260px\|relative angle inequality tetrahedron]]		[[Image:Thefifth.jpg\|thumb\|right\|260px\|relative angle inequality tetrahedron]]
-	For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>x_i</math> and <math>x_j</math> is defined	+	For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>\,x_i\,</math> and <math>\,x_j\,</math> is defined

	<math>\begin{array}{rl}d_{ij}		<math>\begin{array}{rl}d_{ij}
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	Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,		Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,
	for all  <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> ,		for all  <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> ,
-	<math>i\!<\!j\!<\!\ell</math> ,  and for  <math>N\!\geq_{\!}4</math>  distinct points  <math>\{x_k\}</math> ,  the inequalities	+	<math>i\!<\!j\!<\!\ell</math> ,  and for  <math>N\!\geq_{\!}4</math>  distinct points  <math>\,\{x_k\}\,</math> ,  the inequalities

	<math>\begin{array}{cc}		<math>\begin{array}{cc}
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	\end{array}</math>		\end{array}</math>

-	where  <math>\theta_{ikj}\!=_{}\!\theta_{jki}</math>  is the angle between vectors at vertex  <math>x_k</math>  , ~~ ~~ must be satisfied at each point  <math>x_k</math>  regardless of affine dimension.	+	where  <math>\theta_{ikj}\!=_{}\!\theta_{jki}</math>  is the angle between vectors at vertex  <math>\,x_k\,</math> ,  must be satisfied at each point  <math>\,x_k\,</math>  regardless of affine dimension.

	== References ==		== References ==
	* Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry], Meboo, 2007		* Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry], Meboo, 2007

Dattorro at 10:39, 9 November 2007

2007-11-09T10:39:33Z

←Older revision		Revision as of 10:39, 9 November 2007
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		+	[[Image:Thefifth.jpg\|thumb\|right\|260px\|relative angle inequality tetrahedron]]
	For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>x_i</math> and <math>x_j</math> is defined		For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>x_i</math> and <math>x_j</math> is defined

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	==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''==		==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''==
-	[[Image:Thefifth.jpg\|thumb\|right\|260px\|relative angle inequality tetrahedron]]
	Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,		Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,
	for all  <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> ,		for all  <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> ,

Dattorro: /* Fifth property of the Euclidean metric '''('''relative-angle inequality''')''' */

2007-11-09T10:37:53Z

Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''

←Older revision		Revision as of 10:37, 9 November 2007
Line 18:		Line 18:

	==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''==		==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''==
-		+	[[Image:Thefifth.jpg\|thumb\|right\|260px\|relative angle inequality tetrahedron]]
	Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,		Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,
	for all  <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> ,		for all  <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> ,